
The analysis for this system is now standard, which can be found, for example, in [1]}, [2]}, [3]}, [4]} to yield
\(\sum _{|\alpha |\le N}\Vert P_{Pr}\partial ^{\alpha }f\Vert _{L^2_{x,v}} &\le C\sum _{|\alpha |\le N} \left(\Vert \partial ^{\alpha }a\Vert _{L^2_x}+\Vert \partial ^{\alpha }b\Vert _{L^2_x}+\Vert \partial ^{\alpha }c\Vert _{L^2_x}\right) \cr &\le C \sum _{|\alpha |\le N-1}\left( \Vert \partial ^{\alpha }\tilde{l}_0\Vert _{L_{x}^2} + \Vert \partial ^{\alpha }\tilde{h}_0\Vert _{L_{x}^2}\right) \cr &\le C \sum _{|\alpha |\le N}\Vert (I-P_{Pr})\partial ^{\alpha }f\Vert _{L^2_{x,v}} +CM_0 \sum _{|\alpha |\le N}\Vert \partial ^{\alpha }f\Vert _{L^2_{x,v}},\) 
